Heegner Divisors and Nonholomorphic Modular Forms€¦ · modular surface. As an example for Theorem 1.2we then explicitly derive these parts. From there one can obtain the complete

Heegner Divisors and Nonholomorphic

Modular Forms

JENS FUNKE$

Department of Mathematics, Rawles Hall, Indiana University, Bloomington,IN 47405, U.S.A. e-mail: [email protected]

(Received: 16 March 2001; accepted: 6 July 2001)

Abstract. We consider an embedded modular curve in a locally symmetric space M attached

to an orthogonal group of signature ðp; 2Þ and associate to it a nonholomorphic elliptic mod-ular form by integrating a certain theta function over the modular curve. We compute theFourier expansion and identify the generating series of the (suitably defined) intersection num-

bers of the Heegner divisors inM with the modular curve as the holomorphic part of the mod-ular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.

Mathematics Subject Classi¢cations (2000). 11F11, 11F27, 11F30.

Key words. theta series, modular forms, Heegner divisors, intersection numbers.

1. Introduction

In their celebrated paper, Hirzebruch and Zagier [7] show that the intersection num-

bers of certain algebraic cycles on a Hilbert modular surface S occur as the Fourier

coefficients of holomorphic modular forms on the upper half plane H. They expli-

citly compute the intersection numbers of certain curves TN both in S and at the

resolution of the cusp singularities. By direct computation, they then show that these

numbers are Fourier coefficients of modular forms.

These results have inspired numerous other people to look at geodesic cycles in

other locally symmetric spaces and their relationship to modular forms. Here the case

SOð2; qÞwas of particular interest and was studied by Oda [19], Rallis and Schiffmann

[20], Kudla [9] and, recently, by Borcherds [1, 2] and Bruinier [3, 4].

Starting in the late 1970s and throughout the 1980s, Kudla and Millson (see, e.g.,

[14]) carried out an extensive program to explain the work of Hirzebruch–Zagier

from the point of view of Riemannian geometry and the theory of reductive dual

pairs and the theta correspondence. Under some restrictions, they vastly generalize

the results of [7] to orthogonal, unitary, and symplectic groups of arbitrary dimen-

sion and signature. Tong and Wang ([24]) ran a parallel program.

$Fellow of the European PostDoc Institute (EPDI) 99-01.

Compositio Mathematica 133: 289–321, 2002. 289# 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Currently, Kudla, Rapoport and Yang undertake a major investigation of the

occurrence of arithmetic intersection numbers in certain moduli spaces as Fourier

coefficients of modular forms, see, e.g., [11].

This paper deals with the case of the real orthogonal group of signature ðp; 2Þ.

Before we state our results, we need to establish the basic notions of the paper. Let

ðVðQÞ; qÞ be a rational quadratic space of dimension p þ 2 and signature ðp; 2Þ and

let ð ; Þ be the associated nondegenerate symmetric bilinear form on VðQÞ. We have

ðv;wÞ ¼ qðv þ wÞ � qðvÞ � qðwÞ. We let G ¼ SpinðVðQÞÞ viewed as an algebraic group

over Q and write D ¼ GðRÞ=K for the associated symmetric space, where K is a max-

imal compact subgroup of GðRÞ. It is very well known that D is of Hermitian type of

complex dimension p; for example, for p ¼ 1, D ’ H, the upper half plane, and

D ’ H � H for p ¼ 2. We identify D with the space of two-dimensional subspaces

of VðRÞ on which the bilinear form ð ; Þ is negative definite:

D ’ fz VðRÞ : dim z ¼ 2 and ð ; Þjz < 0g: ð1:1Þ

Let L VðQÞ be an integral Z-lattice of full rank, i.e., L L#, the dual

lattice, and G be a congruence subgroup of G preserving L (in the main body of

the paper we will allow a congruence condition as well). We write M ¼ GnD for

the attached locally symmetric space of finite volume. We construct special cyclesin M as follows. Let U V be a positive definite subspace of V of any dimension

04 n4 p. We put

DU ¼ fz 2 D : z U?g; ð1:2Þ

so DU is a complex submanifold of the (same) orthogonal type Oðp � n; 2Þ and codi-

mension n. We can naturally identify SpinðU?Þ ’ GU, where GU is the pointwise sta-

bilizer of U in G. We put GU ¼ G \ GU and define the special cycle

CU ¼ GUnDU: ð1:3Þ

The map CU ! GnD defines an algebraic cycle in M and is actually an embedding if

one passes to a suitable subgroup G0 G of finite index (see [14]). For x 2 VðQÞn, we

denote by UðxÞ the subspace generated by x (of possibly lower dimension) and set

Dx ¼ DUðxÞ and Cx ¼ GxnDx. We will be mainly concerned with the case n ¼ 1, when

the special cycles are divisors. For N 2 N, G acts on LN ¼ fx 2 L : qðxÞ ¼ Ng with

finitely many orbits, and we define the composite cycle

CN ¼X

x2GnLN

Cx: ð1:4Þ

We call CN the Heegnerdivisor of discriminant N. For p ¼ 1, CN is the collection of

Heegner points of discriminant N in a modular (or Shimura) curve, while for p ¼ 2,

CN is a Hirzebruch-Zagier curve TN([7]) in a Hilbert modular surface (if the Q-rank

of G is 1).

Kudla and Millson explicitly construct (in much greater generality) a theta func-

tion yjðt;LÞ ¼P

x2L jðx; tÞ (t ¼ u þ iv 2 H) with values in the closed differential

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ð1; 1Þ-forms of M attached to a certain Schwartz function j ¼ jV on VðRÞ (In Sec-

tion 2, we review this theta function in more detail). They then consider

Ijðt;CÞ ¼

ZC

yjðt;LÞ; ð1:5Þ

the integral of yjðt;LÞ over a compact curve C. Ijðt;CÞ turns out to be a holomorphicmodular form of weight ðp þ 2Þ=2, whose Nth Fourier coefficient is given as the

(cohomological) intersection number of C with the composite cycle CN. This gives

analogues of the original results of Hirzebruch–Zagier in a much more general set-

ting, but actually does not contain their work as they consider the intersection num-

bers of (in general) noncompact cycles. It therefore seems quite naturally to study the

theta integral (1.5) for possibly noncompact curves C. We study the theta integral

Ijðt;CÞ in the noncompact case, where we restrict our attention to the special curves

CU, with U positive definite of dimension p � 1, i.e. to embedded quotients of mod-

ular curves in M. This leads to considerable complications because in [14] the

assumption that C be compactly supported is quite essential and needed at several

places; for example, to guarantee the convergence of the integral (1.5), to show

the holomorphicity in t 2 H, and to verify the vanishing of the negative Fourier

coefficients.

In Section 3 we consider the case p ¼ 1 when M ’ GnH is itself a modular curve,

i.e., V is isotropic, and study the integral Ijðt;MÞ ¼RM yjðt;LÞ. Note that the cusps

of M correspond to the G-equivalence classes of isotropic lines ‘ in V. Our main

result is then that the generating series PðtÞ ¼P1

N¼0 degðCNÞqN of the degree of

Heegner points in M is the holomorphic part of a nonholomorphic modular form

of weight 3=2. Here

degðCNÞ ¼X

x2GnLN

1

jGxjfor N > 0;

while for N ¼ 0 we put degðC0Þ ¼ volðMÞ. More precisely:

THEOREM 1.1.ZM

yjðt;LÞ ¼ PðtÞ þv�1=2

4p

Xcusps ‘

Eð‘;L;GÞXN2Z

bð4pdk2‘N2vÞq�dk2‘N

2

is a nonholomorphic elliptic modular form of weight 3=2. Here Eð‘;L;GÞ denotes the

‘width’ of the cusp ‘ of G ðsee Def: 3.2Þ, d 2 N square-free, the discriminant of the

quadratic space V, k‘ the smallest k 2 N such that L�dk2;‘ ¼ fx 2 L�dk2 : x ? ‘g is

nonempty, and bðyÞ ¼ b3=2ðyÞ ¼R1

1 t�3=2e�ty dt.

For example, specializing to a certain lattice in the quadratic space of discriminant

1, we recover Zagier’s [25] well-known Eisenstein series F of weight 3=2 as a theta

integral. One has

HEEGNER DIVISORS AND NONHOLOMORPHIC MODULAR FORMS 291



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F ðtÞ ¼X1N¼0

HðNÞe2pint þv�1=2

16p

X1N¼�1

bð4pN2vÞe�2piN2t; ð1:6Þ

where HðNÞ denotes the class number of positive definite binary quadratic forms of

discriminant �N2. From this perspective, we can consider Theorem 1.1 on one hand

as a special case of the Siegel–Weil formula expressing the theta integral as an Eisen-

stein series, and on the other hand as the generalization of Zagier’s function to arbi-

trary lattices of signature ð1; 2Þ.

COROLLARY 1.2.RM yjðt;LÞ � 4

P‘

ffiffiffid

pk‘Eð‘;L;GÞF ðdk2‘tÞ is a holomorphic

modular form. Hence, #ðGnLNÞ can be expressed in terms of class numbers and Fouriercoe⁄cients ofa holomorphic modular form.

We also would like to mention the results of [15], where it was shown that the gen-

erating series of the degree of certain 0-cycles in an arithmetic curve (namely, the

moduli scheme of elliptic curves with complex multiplication over the ring of integers

of an imaginary quadratic field) is the holomorphic part of a non-holomorphic mod-

ular form of weight 1. Here the negative Fourier coefficients involve the function

b1ðyÞ ¼R1

1 t�1e�tydt. The similarities of the situation over C and over number fields

are quite striking. We also compute the Mellin transform LðsÞ ofRM yjðt;LÞ. It turns

out to be closely related to certain Siegel zeta functions associated to (split) indefinite

quadratic forms of signature ð1; 2Þ which were previously studied by Shintani and

F. Sato (see [22, 21]). We have

THEOREM 1.3.

LðsÞ ¼ ð2pÞ�sGðsÞz2ðs;LÞ þ 2�1�sp�12�sG s � 1

2

� � 1sF 3

2; s; s þ 1;�1� �

z1ðs;LÞ;

where F ¼ 2F1 is the hypergeometric function, and the Siegel zeta functions are defined

by

z1ðs;LÞ ¼Xx2L

x? split

jqðxÞj�s and z2ðs;LÞ ¼Xx2L

qðxÞ>0

1

jGxjjqðxÞj�s:

The proof of Theorem 1:1 is long and occupies Section 4. After showing the con-

vergence ofRM yjðt;LÞ using a Poisson summation argument, we are reduced to cal-

culating the orbital integralsZGnD

Xg2GxnG

g�jðxÞ ð1:7Þ

for elliptic, hyperbolic, and parabolic stabilizer Gx. (In the parabolic case, Gx shall

mean here the stabilizer of the isotropic line generated by x, and one first has to

sum over all isotropic vectors before integrating.)

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We treat the theta integral ð1:5Þ for general p and a modular curve C ¼ CU in

Section 5. The result is

THEOREM 1.4. LetU Vbe positive de¢nite ofdimension p � 1 so thatCU ’ GUnH.Assume for simplicity L ¼ L \ U þ L \ U?. Then

RCU

yjVðt;LÞ is nonholomorphic for

CU noncompact andZCU

yjVðt;LÞ ¼ yðt;L \ UÞ

ZCU

yjU?ðt;L \ U?Þ;

where yðt;L \ UÞ ¼P

x2L\U e2piqðxÞt is the standard theta series attached to the positive

definite lattice L \ U and the second integral is the one considered in Th. 1:4 for the

space U?, which has signature ð1; 2Þ. Moreover, the holomorphic Fourier coefficients

can be interpreted as the intersection numbers in ðthe interior of Þ M of the curve CU

and the divisor CN.

This generalizes parts of the results of Hirzebruch–Zagier, namely the ones con-

cerning the intersection numbers of the curves TN in the ‘interior’ of the Hilbert

modular surface. As an example for Theorem 1.2 we then explicitly derive these

parts. From there one can obtain the complete result of Hirzebruch–Zagier by

applying the holomorphic projection principle for modular forms to the theta

integral. This will then account for the contribution of the cusps to the intersec-

tion numbers (This is an idea of van der Geer and (independently) Zagier; see

[23].) However, when using this procedure, one still needs explicit formulas for

the intersection at the cusps. As this seems infeasible for the higher dimensional

case, a more conceptual approach for the cusps (similar to the treatment of the

‘interior’ presented here) is still needed. Our results should be closely related to

recent work of Borcherds [2] and Bruinier [3, 4]. They showed that the generating

seriesP1

N¼0 CNqN of Heegner divisors, when considered as elements in the so-

called ‘Heegner divisor class group’, is a holomorphic modular form of weight

ðp þ 2Þ=2 with values in this group in the sense that application of a linear form

on this Heegner divisor class group gives a scalar-valued holomorphic modular

form. One should expect that for p5 2, an extension of the methods used here

to the cusps should yield the results of [2, 3, 4] (while it seems that the methods

used there will not imply our results). For p ¼ 1, the results actually

are independent of each other, as taking the degree is the zero map on the

‘Heegner divisor class group’.

2. Work of Kudla and Millson

Kudla and Millson explicitly construct for orthogonal, unitary, and symplectic

groups of arbitrary signature the Poincare dual form of the cycles CU ([12, 13]).

We denote by Or;sðDÞ the space of smooth differentials forms of type ðr; sÞ on D.




https://doi.org/10.1023/A:1020002121978


THEOREM 2.1 ([12, 13]). For each n with 04 n4 p, there is a nonzero Schwartz form

jðnÞ 2 SðVðRÞnÞ � On;n

ðDÞ½ �G

ð2:1Þ

such that

ðiÞ djðnÞ ¼ 0; i.e., for each x 2 Vn, jðnÞðxÞ is a closed ðn; nÞ-form on D which is Gx-invariant: g�jðnÞðxÞ ¼ jðnÞðxÞ for g 2 Gx; the stabilizerof x inG.

ðiiÞ Denote by jþUðxÞ ¼ e�ptrðx;xÞ with x 2 Un and ðx; xÞi;j ¼ ðxi; xjÞ the standard Gaus-

sian on a positive de¢nite subspace U of VðRÞ. Then, under the pullbacki�U: On;n

ðDÞ ! On;nðDUÞ ofdi¡erential forms, we have

i�UjðnÞ ¼ jþ

U � jðnÞU? ;

where jðnÞU? is the nth Schwartz form for U? and GU.

ðiiiÞ Assume U ¼ UðxÞ for a linear independent n-frame in U.Then the Poincare¤ dual ofGUnDU is given by

epðx;xÞX

g2GUnG

g�jðnÞðxÞ

" #:

From now on we will restrict our attention to the case n ¼ 1. For simplicity we will

write j for the ð1; 1Þ-form jð1Þ.

Remark 2.2. We do not need the explicit general formula for the Schwartz form

right now; for an easily accessible construction see [9]. For signature ð1; 2Þ, we will

give the formula in the next section.

We denote by gSL2ðRÞ the two-fold cover of SL2ðRÞ. Recall that gSL2ðRÞ acts on the

Schwartz space SðVðRÞÞ via the Weil representation o associated to the additive

character t j�! expð2pitÞ, see for example [17]. Let K0 gSL2ðRÞ be the inverse image

of the standard maximal subgroup SOð2Þ in SL2ðRÞ.

PROPOSITION 2.3 ([12]). The Schwartz form j is an eigenfunction for K0 withrespect to theWeil representation i.e.,

oðk0Þj ¼ detðk0Þðpþ2Þ=2j: ð2:6Þ

We associate to j in the usual way a function on the upper half plane H. For

t ¼ u þ iv 2 H we put

g0t ¼

1 u0 1

� �v1=2 00 v�1=2

� �; ð2:7Þ

and define

jðt; xÞ ¼ v�ðpþ2Þ=4oðg0tÞjðxÞ: ð2:8Þ

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Fix a congruence condition h 2 L# and assume that G fixes the coset L þ h under

the action on L#=L. Then the theta kernel

yjðtÞ ¼ yjðt;L; hÞ ¼X

x2hþL

jðt; xÞ 2 O1;1ðDÞ

Gð2:9Þ

defines a closed ð1; 1Þ-form inM ¼ GnD. By the usual theta machinery (Poisson sum-

mation) it is a nonholomorphic modular form for the congruence subgroup GðNÞ of

SL2ðZÞ (where N is the level of the lattice L), of weight ðp þ 2Þ=2 with values in

O1;1ðDÞ

G. Slightly more general than in the introduction we put

Lm ¼ fx 2 L þ h : qðxÞ ¼ mg for m 2 Q;

suppressing the dependence on h. For m > 0, we again obtain a composite divisor

Cm ¼P

x2GnLmCx in M. Let Z be a closed ðp � 1; p � 1Þ-form on M and assume that

Z is rapidly decreasing if M is noncompact. The main result of [14] (in much greater

generality) is that

Ijðt; ZÞ ¼

ZM

yjðtÞ ^ Z ð2:10Þ

is a holomorphic modular form whose Fourier coefficients are periods of Z over the

composite cycles Cm in M. If Z now represents the Poincare dual class of a compactcurve C in M, we obtain

Ijðt;CÞ ¼

ZC

yjðtÞ ¼ Ijðt; ZÞ; ð2:11Þ

and the Fourier coefficients are the (cohomological) intersection numbers of C with

Cm. In the following we consider (2.11) for modular curves C.

3. The Theta Integral Associated to SO ð1; 2Þ

3.1. PRELIMINARIES

Now assume dimV ¼ 3; hence V has signature ð1; 2Þ. Over R we fix an isomorphism

VðRÞ ’x1 x2x3 �x1

� �2 M2ðRÞ

ð3:1Þ

such that

qðXÞ ¼ detðXÞ ¼ �x21 � x2x3 and ðX;YÞ ¼ �trðXYÞ:

So we can view VðRÞ as the trace zero part B0ðRÞ of the indefinite quaternion algebra

BðRÞ ¼ M2ðRÞ over R. We have G ¼ SpinðVÞ ¼ SL2 and the action on B0 is the con-

jugation:

g � X :¼ gXg�1 ð3:2Þ

for X 2 B0 and g 2 G.




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Notation. In this section, we will write z ¼ x þ iy for an element in D ’ H (the

orthogonal variable) and t ¼ u þ iv 2 H for the symplectic variable. The upper case

letters X and Y we reserve for vectors in VðRÞ with coefficients xi and yi.

Here it is more convenient to consider the symmetric space D ’ H not as the space

of negative two-planes in VðRÞ but rather as the space of positive lines. We give the

following identification with the upper half plane. Picking as base point of D the line

z0 spanned by�0 1�1 0

�, we note that K ¼ SOð2Þ is its stabilizer in GðRÞ, and we have

the isomorphism:

H ’ GðRÞ=K ! D ð3:3Þ

with

z j�! gK j�! g � z0 ¼: ‘ðzÞ ð3:4Þ

(where g 2 GðRÞ such that gi ¼ z; the action is the usual linear fractional transforma-

tion). We find that ‘ðzÞ is generated by

XðzÞ :¼ y�1 � 12 ðz þ �zÞ z �z�1 1

2 ðz þ �zÞ

� �; ð3:5Þ

where z ¼ x þ iy. Note qðXðzÞÞ ¼ 1. Moreover, per construction

g � XðzÞ ¼ XðgzÞ: ð3:6Þ

For X ¼�x1 x2x3 �x1

�2 B0ðRÞ we compute

ðX;XðzÞÞ ¼ �y�1ðx3z �z � x1ðz þ �zÞ � x2Þ

¼ �y�1ðx3ðx2 þ y2Þ � 2x1x � x2Þ

¼ðx3x � x1Þ

2þ qðXÞ

�x3y� x3y; ð3:7Þ

when x3 6¼ 0.

The minimal majorant of ð ; Þ associated to z 2 D is given by

ðX;XÞz ¼ðX;XÞ; if X 2 ‘ðzÞ;

�ðX;XÞ; if X 2 ‘ðzÞ?:

ð3:8Þ

A little calculation shows

ðX;XÞz ¼ ðX;XðzÞÞ2 � ðX;XÞ

¼ðx3x

2 � x2 � 2x1xÞ2

y2þ ðx3yÞ

2þ 2ðx3x � x1Þ

2: ð3:9Þ

PROPOSITION 3.1 ([10]). The Schwartz function j ¼ jð1Þ on VðRÞ valued in theð1; 1Þ-forms onD is explicitly given by

jðX; zÞ ¼

�ðX;XðzÞÞ2 �

1

2p

�e�pðX;XÞz o: ð3:10Þ

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Here

o ¼dx ^ dy

y2¼

i

2

dz ^ d �z

y2; ð3:11Þ

the standard G-invariant ð1; 1Þ-form on D ’ H.

We will write jðXÞ for its value at X. Then

g�jðXÞ ¼ jðg�1XÞ; ð3:12Þ

i.e., jðg � X; gzÞ ¼ jðX; zÞ for g 2 GðRÞ, follows from (3.6); while Proposition 2.3

becomes an exercise using the explicit formulae of the Weil representation and redu-

ces to j ¼ �j. Finally, we define

j0ðX; zÞ ¼ epðX;XÞjðX; zÞ ð3:13Þ

¼ ðX;XðzÞÞ2 �1

2p

� �e�pðX;XðzÞÞ2þ2pðX;XÞ o ð3:14Þ

and

jðt;XÞ ¼

�vðX;XðzÞÞ2 �

1

2p

�epiðX;XÞz;t o; ð3:15Þ

where

ðX;XÞz;t ¼ uðX;XÞ þ ivðX;XÞz:

¼ �tðX;XÞ þ ivðX;XðzÞÞ2: ð3:16Þ

3.2. THE THETA INTEGRAL

We put

IjðtÞ :¼ZGnD

yðt;L; hÞ ¼

ZGnD

XX2Lþh

jðt;XÞ: ð3:17Þ

Thus, in the notation of the previous section, IjðtÞ ¼ yjðt; ZÞ where Z is the constant

function 1. In particular, Z is not rapidly decreasing. Alternatively we can interpret

(3.21) as the integral Ijðt;CUÞ in the case of signature ð1; 2Þ with U ¼ 0. We assume

the convergence of (3.17) for the moment. Then it is again clear that IðtÞ defines a (in

general nonholomorphic) modular form on the upper half plane of weight 3=2. For

X 2 Lm, we have

jðt;XÞ ¼ qmj0ðffiffiffiv

pXÞ; ð3:18Þ

where qm ¼ e2pimt, as usual. Define

ymðtÞ ¼XX2Lm

jðt;XÞ and y0mðvÞ ¼XX2Lm

j0ðffiffiffiv

pXÞ: ð3:19Þ




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We then – yet formally – have

IjðtÞ ¼

ZGnD

Xm2Q

ymðtÞ ¼Xm2Q

ZGnD

y0mðvÞ

� �qm; ð3:20Þ

which is the Fourier expansion of IjðtÞ. Since GnLm is finite for m 6¼ 0, we can

simplify the inner integral in (3.20) in that case:ZGnD

y0mðvÞ ¼

ZGnD

XX2GnLm

Xg2GXnG

j0ðg�1ffiffiffiv

pX; zÞ

¼X

X2GnLm

ZGnD

Xg2GXnG

g�j0ðffiffiffiv

pX; zÞ: ð3:21Þ

For X ¼ 0, we simply have jðt;XÞ ¼ �ð1=2pÞo and thereforeZGnD

jðt; 0Þ ¼ mðGnDÞ; ð3:22Þ

the hyperbolic volume of GnD, normalized such that mðSL2ðZÞnHÞ ¼ � 16. In particu-

lar, we see that with this normalization (see, e.g., [18])

mðGnDÞ 2 Q: ð3:23Þ

If V is isotropic over Q; we can pick the isomorphism (3.1) such that

VðQÞ ’

ffiffiffid

px1 x2

x3 �ffiffiffid

px1

� �: xi 2 Q

¼: B0ðd;QÞ ð3:24Þ

as quadratic Q-vector spaces, where d is a square-free positive integer, the discrimi-

nant of the quadratic space V. The set of all isotropic lines in V corresponds to the

cusps of GðQÞ and G acts on them with finitely many orbits. For the constant term in

the Fourier expansion of IjðtÞ, we therefore have to proceed differently than in the

case m 6¼ 0: Let ‘1; . . . ; ‘t be a set of G-representatives of isotropic lines and pick

Xi 2 ‘i primitive in L. Define dð‘iÞ ¼ dð‘i;L; h;GÞ ¼ 1 if ‘i intersects the coset

L þ h and dð‘iÞ ¼ 0 otherwise. Hence, ‘i \ ðL þ hÞ ¼ ZXi þ hi for some hi 2 ‘i if

dð‘iÞ ¼ 1. Denote by Gi the stabilizer of ‘i in G. We then haveZGnD

XX2L0X 6¼0

j0ðffiffiffiv

pX; zÞ ¼

ZGnD

Xti¼1

XX2Gð‘i\ðLþhÞÞ

X6¼0

j0ðffiffiffiv

pX; zÞ

¼

ZGnD

Xti¼1

XX2‘i\ðLþhÞ

X 6¼0

Xg2GinG

g�j0ðffiffiffiv

pX; zÞ

¼Xti¼1

dð‘iÞZGnD

Xg2GinG

X1k¼�1

0

g�j0ðffiffiffiv

pðkXi þ hiÞ; zÞ: ð3:25Þ

HereP0 indicates that we omit k ¼ 0 in the sum in the case of the trivial coset. We

need to discuss the notion of the width of a cusp for our purposes. For Y 2 V

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isotropic (usually primitive in L), pick g 2 GðRÞ such that gY ¼ bX0 with

X0 ¼�0 10 0

�and b 2 R�. Put G0 ¼ gGg�1. Hence, G0

X0¼ gGYg

�1 is equal to��1 ka0 1

�: k 2 Z

�(if �I 2 G) for some a 2 Rþ. Now we could call a the width

of the cusp ‘, however this is not well defined, since it depends on the choice of

g 2 GðRÞ and, hence, on b. However, one easily checks that the ratio a=jbj is constantand only depends on Y and G.

DEFINITION 3.2. (Width of a cusp). (i) For Y isotropic, we define EðY;GÞ ¼ a=jbj;where a and b are the quantities above. We call the pair ðY;GÞ a cusp and the number

EðY;GÞ its width.

(ii) For a lattice L V, h 2 L#=L and G GðLÞ, we define (with the above nota-

tion) the total width by EðL; h;GÞ ¼Pt

i¼1 dð‘iÞEðXi;GÞ:

Remark 3.3. (i) If G is a congruence subgroup of SL2ðZÞ, then there is (up to sign)

a unique g 2 SL2ðZÞ such that gY ¼ bX0 ¼ b�0 1

0 0

�. Then both numbers a and b are

intrinsic: a is simply the usual width of a cusp of a congruence subgroup of SL2ðZÞ

while b can be interpreted as the volume of the fundamental domain of RX0=ZbX0

with respect to the Lebesgue measure on RX0.

(ii) We can always arrange jbj ¼ 1. Then we can interpret a ¼ EðY;GÞ as the

volume of the corresponding component of the Borel–Serre boundary of M.

(iii) It does indeed happen that for a fixed space V we can find two lattices with the

same stabilizer G such that the ’cusp 1’ has different width. Consider the lattices

b ac �b

� : a; b; c 2 Z

n oand b 2a

2c �b

� : a; b; c 2 Z

n o:

Both have stabilizer G ¼ SL2ðZÞ, hence a ¼ 1, but b ¼ 1 and 2, respectively. See also

Example 3.9.

We are now ready for the main result:

THEOREM 3.4.With the above notationwe have

ðiÞ yjðtÞ 2 L1ðGnDÞ;

ðiiÞ y0mðvÞ 2 L1ðGnDÞ;

forallm 2 QandZGnD

yjðtÞ ¼Xm2Q

ZGnD

y0mðvÞ

� �qm:

For the Fourier coefficients; we get

ðiiiÞ form > 0:ZGnD

y0mðvÞ ¼X

X2GnLm

1

jGXj;




https://doi.org/10.1023/A:1020002121978


ðivÞ form ¼ 0:ZGnD

y0mðvÞ ¼ mðGnDÞ þ1

2pv�1=2EðL; h;GÞ;

where the volume term only occurs for h 2 L;(v) form < 0:Z

GnD

y0mðvÞ ¼1

4pffiffiffiffiffiffiffiffi�m

p v�1=2X

X2GnLmX? isotropic

Z 1

1

t�3=2e4pvmtdt

� �:

Note that for m < 0 we haveZ 1

1

t�3=2e4pmtdt4Oðe4pmÞ ð3:26Þ

so that the negative part of the Fourier expansion has the same convergence

behavior as the positive part. The next section will deal with the proof of the

Theorem 3.4.

First we discuss the convergence of IjðtÞ. Then we turn our attention to the com-

putation of the individual integrals which define the Fourier coefficients via the ana-

lysis (3.19) to (3.25). For m > 0, the composite 0-cycle Cm ¼P

X2GnLmCX is the

collection of Heegner points of discriminant m, and we define its degree by

degðCmÞ ¼X

X2GnLm

1

jGXj: ð3:27Þ

We also put

degðC0Þ ¼mðGnDÞ; if 0 2 L þ h,0; else.

ð3:28Þ

In any case we have degðCmÞ 2 Q. We define the generating series of the degree of the

Heegner points by

PðtÞ ¼Xm5 0

degðCmÞqm: ð3:29Þ

Then as a corollary we obtain the following generalization of the results of Kudla

and Millson to the noncompact case:

THEOREM 3.5.LetVbe a quadratic space over Qof signature ð1; 2Þ.

ðiÞ ð½14�Þ If V is anisotropic, i.e., GnD compact, then the generating function PðtÞ is aholomorphic modular form of weight 3=2 for a suitable congruence subgroup ofSL2ðZÞ.

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ðiiÞ If V is isotropic, i.e., GnD noncompact, then PðtÞ is the holomorphic part of the non-holomorphic modular form of weight 3=2given byXm5 0

degðCmÞqm þv�1=2

2pEðL; h;GÞ þ

Xm>0

v�1=2

4pffiffiffiffim

pX

X2GnL�mX? isotropic

bð4pvmÞ q�m; ð3:30Þ

where we set; following Zagier ½25� ðup to a factorÞ, bðsÞ ¼R1

1 t�3=2e�stdt.

The following lemma characterizes the values for which the negative Fourier coef-

ficients in the previous theorem are nonzero.

LEMMA 3.6. For X 2 VðQÞ ’ B0ðd;QÞ with qðXÞ < 0 the following two statements areequivalent:

ð1Þ X? is split over Q.ð2Þ qðXÞ 2 �d Q

�ð Þ

2.

Proof. If qðXÞ ¼ �dm2 < 0 with m 2 Q, then by Witt’s Theorem we can map

X 7!�m

ffiffid

p0

0 �mffiffid

p�

2 B0ðd;QÞ; hence X? is split. Conversely, if X? is split, we can

assume X ?�0 1

0 0

�(again by Witt’s Theorem moving an isotropic vector orthogonal

to X to�0 1

0 0

�. Thus X ¼

�m

ffiffid

p�

0 �mffiffid

p�for some m 2 Q. &

Let X 2 L�dm2 . Then X is orthogonal to two cusps. However, giving an orientation

to QX we can distinguish between these cusps (if they are not equivalent), since

switching the cusps by an element in G \ SOðX?Þ switches X to �X. For a fixed cusp

‘i, we write

L�dm2;i;þ ¼ fX 2 L�dm2 : X ? ‘i; X pos orient:g

and note that Gi acts on this set. We have

LEMMA 3.7.

#GnL�dm2 ¼Xti¼1

#GinL�dm2;i;þ

and

#GinL�dm2;i;þ ¼ 2mffiffiffid

pEðXi;GÞ;

if L�dm2;i;þ is not empty.

Proof. The first assertion is clear. For the second, take X 2 L�dm2 , say

X ¼�m

ffiffid

p0

0 �mffiffid

p�. So X is orthogonal to the cusps 0 and 1. We distinguish them by

requiring that the left upper left entry of X is positive, since switching the cusps by�0 1

�1 0

�maps X to �X. By our conventions about the cusps we can assume that�

0 b0 0

�is primitive in L with stabilizer




https://doi.org/10.1023/A:1020002121978


G1ðaÞ ¼ Tak ¼ �1 ak0 1

� �: k 2 Z

in G ðif � I 2 GÞ:

Now

Tak � X ¼m

ffiffiffid

p�2m

ffiffiffid

pak

0 �mffiffiffid

p

� �; ð3:31Þ

hence

X � Tak � X ¼0 �2m

ffiffiffid

pak

0 0

� �2 L: ð3:32Þ

Thus 2mffiffiffid

pa 2 bZ. The assertion now follows from the observation that we have

showed that we have 2mffiffiffid

pE equivalence classes of vectors in L�dm2;i;þ. &

Lemmata 3.6 and 3.7 enable us to rewrite Theorem 3.5(ii):

THEOREM 3.8. With the above notationwe have

Ijðt;L; hÞ ¼Xm�0

degðCmÞqm þv�

12

2p

Xcusps ‘i

EðXi;GÞ�

� dð‘iÞ þXm2Qþ

L�dm2;i;þ6¼;

bð4pdm2vÞq�dm2

0BBBBBB@

1CCCCCCA:

This implies Theorem 1.1 as follows: For the constant Fourier coefficient note

that for the trivial coset we have dð‘iÞ ¼ 1 and bð0Þ ¼ 2. For the negative coeffi-

cients, take a nonisotropic vector X in ‘?i , primitive in L. We see qðXÞ ¼ �dk2‘i

for some k‘i 2 N. Now all other vectors in the sum for the cusp ‘i are integral mul-

tiples of X.

EXAMPLE 3.9 (Zagier’s Eisenstein series of weight 3=2). Let V be the space of trace

0 elements of the (indefinite) split quaternion algebra M2ðQÞ over Q; i.e.,

V ’ B0ð1;QÞ. (i). We first consider the isotropic lattice

L ¼b 2a2c �b

� �: a; b; c 2 Z

; ð3:33Þ

so qða; b; cÞ ¼ �b2 � 4ac. L has level 4. As mentioned above, we have G ¼ GðLÞ ¼

SL2ðZÞ and one isomorphism class of cusps. The stabilizer of ‘0 ¼ Q�0 2

0 0

�is

G1 ¼ f��1 n

0 1

�: n 2 Zg. Note that the width of this cusp (in the above sense) is

E ¼ 12! By Lemma 3.7 G acts on L�n2 with n orbits, and these X are precisely the

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vectors such that X? is split over Q. For N > 0 and X ¼�

b 2a

2c �b

�such that

qðXÞ ¼ N, observe that the assignment

X j�!�2c bb 2a

� �ð3:34Þ

defines (for a > 0) a binary positive definite quadratic form of discriminant �N. One

easily deduces that

# GnLNð Þ ¼ 2HðNÞ; ð3:35Þ

where HðNÞ denotes the class number of binary positive definite integral quadratic

forms of discriminant �N; we count the classes with nontrivial stabilizer with multi-

plicities 1=2 and 1=3, respectively. The elements of the form k�

0 2

�2 0

�(corresponding

to i 2 H) are fixed by�

0 1

�1 0

�and the stabilizer of the elements of the form k

�1 2

�2 1

�(corresponding to ð1 þ i

ffiffiffi3

pÞ=2) is generated by

�1 �1

1 0

�. We set Hð0Þ ¼ � 1

12 ¼12 mðGnDÞ. Applying Theorem 1.1 we obtain

1

2Ijðt;LÞ ¼ F ðtÞ ¼

X1N¼0

HðNÞqN þv�1=2

16p

Xn2Z

bð4pn2vÞ q�n2 ; ð3:36Þ

Zagier’s well known nonholomorphic Eisenstein series of weight 3=2 and level 4,

see [25,7]. (ii). This example we will need later to deduce a case of the results of

Hirzebruch and Zagier. We almost repeat (i) considering the trace zero elements

in the maximal order M2ðZÞ in M2ðQÞ:

K ¼b ac �b

� �: a; b; c 2 Z

; ð3:37Þ

so qða; b; cÞ ¼ �b2 � ac. We set X1 ¼�1 0

0 �1

�and define (the coset)

Kj ¼j

2X1 þ K ð3:38Þ

for j ¼ 0; 1. Again we put G ¼ SL2ðZÞ ¼ GðKjÞ. This time we have E ¼ 1 and d ¼ 0

for j ¼ 1. As above we see that G acts on the vectors of length �ðn þ ðj=2ÞÞ2 with

2n þ j orbits. For N > 0 and X ¼�bþj=2 a

c �b�j=2

�such that qðXÞ ¼ N �

j4 we assign

X j�!�2c 2b þ j2b þ j 2a

� �; ð3:39Þ

which defines a positive form of discriminant �4N þ j! We obtain

#ðGnðKjÞN�j4Þ ¼ 2Hð4N � jÞ; ð3:40Þ

hence

1

2Ijðt;KjÞ ¼

X1N¼0

Hð4N � jÞqN�j=4 þv�1=2

8p

Xn2Z

b 4p n þj

2

� �2

v

!q�ðnþ j

2Þ2

: ð3:41Þ




https://doi.org/10.1023/A:1020002121978


The example now implies Corollary 1.2.

As a corollary to the example and as an illustration of Corollary 1.2 we obtain the

famous relationship between the representation numbers of integers as a sum of

three squares and the class numbers of binary quadratic forms:

COROLLARY 3.10. Denote by r3ðNÞ the representation number of N as a sum of

three squares and write WðtÞ ¼P

n2Z e2pin2t for the classical Jacobi theta series. Then

WðtÞð Þ3¼ 12

1

2Ijðt;K0Þ � Ijðt;LÞ

� �;

i.e.,

r3ðNÞ ¼ 12 Hð4NÞ � 2HðNÞð Þ

for all integers N.

Proof. The space of holomorphic modular forms for G0ð4Þ of weight 3=2 of

Nebentypus is one-dimensional, spanned by WðtÞð Þ3. &

We now compute the Mellin transform LðsÞ of IjðtÞ, hence giving the proof of

Theorem 1.3: We write

LðsÞ ¼ LþðsÞ þ L�ðsÞ ð3:42Þ

with

L�ðsÞ ¼

Z 1

0

XX2Lþh�qðXÞ>0

jðiv;XÞvsdv

v: ð3:43Þ

For the positive part, we obtain in the standard fashion via Theorem 3.4 (iii)

LþðsÞ ¼ ð2pÞ�sGðsÞX

X2LþhqðXÞ>0

1

jGxjqðXÞ

�s¼ ð2pÞ�sGðsÞz2ðs;L; hÞ: ð3:44Þ

For the negative part, we have via Theorem 3.4 (v)

L�ðsÞ ¼

Z 1

0

XX2LþhX? split

v�12

4pffiffiffiffiffiffiffiffiffiffiffiffijqðXÞj

p Z 1

1

u�32e�4pvjqðXÞjudu

� �e2pjqðXÞjvvs

dv

vð3:45Þ

¼X

X2LþhX? split

v�12


p Z 1

1

Z 1

0

e�2pjqðXÞjð2u�1Þvvs�12dv

v

� �u�3

2du ð3:46Þ

¼X

X2LþhX? split

v�12


p G s � 12

� � Z 1

1

ð2pjqðXÞjð2u � 1ÞÞ12�su�3

2du ð3:47Þ

¼ 2�32�sp�1

2�sG s � 12

� �z1ðs;L; hÞ

Z 1

1

ð2u � 1Þ�12�s

u3=2du: ð3:48Þ

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The integral in the last line is equal to

ffiffiffi2

pZ 1

0

ws�1

ðw þ 1Þ3=2¼

ffiffiffi2

p 1

sF 3

2

� �; s; s þ 1;�1Þ;

see [16]. This concludes the proof of Theorem 1.3.

4. Proof of Theorem 3.4

4.1. CONVERGENCE OF THE THETA INTEGRAL

PROPOSITION 4.1 (Theorem 3.4 (i)). yjðt;L; hÞ 2 L1ðGnDÞ:

Proof. If GnD is compact, i.e., V ¼ VðQÞ is anisotropic, then there is nothing to

show; the existence of the integral is immediate. On the other hand, if V is isotropic

then we can choose the isomorphism of VðRÞ ’ B0ðRÞ such that VðQÞ ’ B0ðd;QÞ

and X0 :¼�0 1

0 0

�primitive in L. We then can find r 2 Q such that

L0 :¼ rB0ðd;ZÞ L. With this notation we see

yjðt;L; hÞ ¼X

h0�hðLÞmodL0

yjðt;L0; h0Þ: ð4:1Þ

So it is sufficient to show that each yjðt;L0; h0Þ 2 L1ðGnDÞ. Picking a fundamental

domain for GnD we observe that via (3.12) it suffices to show that yj is rapidly

decreasing as y ! 1. For X 2 B0ðd;QÞ, we have

jðt;XÞ ¼

�vðX;XðzÞÞ2 �

1

2p

�e�pvðX;XðzÞÞ2e2pi�tqðXÞo

¼v

y2ðx3z �z � 2

ffiffiffid

px1x � x2Þ

2�

1

2p

� ��

� exp �pv

y2ðx3z �z � 2

ffiffiffid

px1x � x2Þ

2

� ��

� exp ��tðdx21 þ x2x3Þ� �

o: ð4:2Þ

Write x2 ¼ x02 þ h0

2 and let x02 run over rZ. We will apply Poisson summation to the

sum on x02. So consider in the above expression the coefficient of o as a function

f of x02. For the Fourier transform of f, we see by changing variables to

�t ¼

ffiffiffiv

p

yðx3z �z � 2

ffiffiffid

px1x � x0

2 � h02Þ;




https://doi.org/10.1023/A:1020002121978


fðwÞ ¼

Z 1

�1

fðx02Þ exp ð�2pix0

2wÞdx02

¼ ðyffiffiffiv

pÞ�1 exp

��tdx21

�expð�½w þ x3 �t�½x3z �z � 2

ffiffiffid

px1x � h0

2�Þ�

�

Z 1

�1

t2 �1

2p

� �expð�pt2Þ exp �t

yffiffiffiv

p ðw þ x3 �tÞ� �

dt

¼ �ð�1Þyffiffiffiv

p ðw þ x3 �tÞ� �2

exp �pyffiffiffiv

p ðw þ x3 �tÞ� �2

!; ð4:3Þ

since the Fourier transform of ðt2 � ð1=2pÞÞ expð�pt2Þ is �t2 exp ð�pt2Þ. We obtain:

yðt;L0; h0ÞðzÞ ¼ �r�1 y

v3=2

Xw2r�1Z

x12h01þrZ

x32h03þrZ

ðw þ x3 �tÞ2 expð��tdx21Þ�

� expð�½w þ x3 �t�½x3z �z � 2ffiffiffid

px1x � h0

2�Þ exp �py2

vðw þ x3 �tÞ

2

� �o:

All terms of this sum are exponentially decreasing as y 7!1 except those for which

w ¼ x3 ¼ 0. But the coefficient w þ x3 �tð Þ2 vanishes for such terms. So jyðt;L0; h0Þj is

integrable and yðt;L; hÞ is real analytic in t. &

4.2. COMPUTATION OF THE INDIVIDUAL TERMS

We will compute for G an arbitrary Fuchsian subgroup of SL2ðRÞ the integralsZGnD

Xg2GXnG

g�j0ðX; zÞ ð4:4Þ

with any nonisotropic X 2 VðRÞ and GX the stabilizer of X in G, andZGnD

XX2GðZYþhYÞ

X6¼0

j0ðX; zÞ; ð4:5Þ

where Y 2 VðRÞ isotropic such that QY ¼ ‘ is a cusp of G, i.e., GY is nontrivial, and

hY 2 ‘. Note that via (3.21) and (3.25) the calculation of these integrals yields the

Fourier expansion of IjðtÞ (Theorem 3.4).

There are several cases to consider:

(A) qðXÞ > 0;

(B1) qðXÞ < 0, GX nontrivial, infinite cyclic,

(B2) qðXÞ < 0, GX trivial,

(C) The integral (4.5).

According to the stabilizer GX of X (G‘ of the isotropic line ‘) we call (A) the

elliptic, (B) the hyperbolic and (C) the parabolic case. Note that the cases are closely

related to each other:

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LEMMA 4.2. Let qðXÞ < 0 forX 2 VðQÞ, soX? has signature ð1; 1Þ.ThenGX is trivial ifX? splits over Q. Conversely, if X? is nonsplit, i.e., anisotropic over Q, then GX is in¢nitecyclic.

Proof. Indeed, we can identify GXðRÞ with SpinðX?Þ ’ R� which acts on the two

isotropic lines of X? as homotheties. Hence, GX ¼ GðX?Þ being a discrete subgroup

of R� is either trivial or infinite cyclic. But for a possible isotropic line ‘ over Q, i.e.,

a cusp for G, we would also have a discrete unipotent subgroup in G stabilizing ‘.

This together with GðX?Þ has an accumulation point, see [18], p. 16. On the other

hand, an indefinite anisotropic binary quadratic form over Q corresponds to the

norm form of a real quadratic field K over Q, and the units O�K act as isometries and

are infinite cyclic (up to torsion). &

So the cases (B2) and (C) occur together. Moreover, they occur if and only if GnD

is noncompact. These two cases create the complications when extending the results

of Kudla and Millson to the noncompact case.

4.2.1(A). The Elliptic Case

Let X 2 VðRÞ such that qðXÞ > 0, hence RX 2 D. Therefore the stabilizer GX of X in

GðRÞ ¼ SL2ðRÞ is conjugate to SO2ðRÞ, and is in particular compact. Since GX is a

discrete subgroup, it is finite cyclic.

PROPOSITION 4.3 (Theorem 3.4 (iii)). Let qðXÞ > 0.ThenXg2GXnG

jg�jðX; zÞj 2 L1ðGnDÞ;

unfolding in ð4:4Þ is allowed, and we haveZGnD

Xg2G

g�j0ðX; zÞ ¼

ZD

j0ðX; zÞ ¼ 1: ð4:6Þ

Remark 4.4. This result is certainly already contained in [12,13]. In fact, it is one of

the corner stones of the theory. However, for the convenience of the reader we give a

brief sketch of the proof.

Proof. Write X ¼�x1 x2x3 �x1

�. Since qðXÞ > 0, we have x3 6¼ 0. Hence by the explicit

formulae for j we get

j0ðX; zÞ ¼ e2pðX;XÞ ðx3x � x1Þ2þ qðXÞ

�x3y� x3y

� �2

�1

2p

" #�

� e�p

ðx3x�x1 Þ2þqðXÞ

�x3y�x3y

� 2

dxdy

y2: ð4:7Þ




https://doi.org/10.1023/A:1020002121978


The integrand is rapidly decreasing for x ! �1 and y ! 0;1. Note that here one

needs qðXÞ > 0. Hence unfolding is allowed and therefore j0ðX; zÞ 2 L1ðDÞ. The

proof ofRD j0ðX; zÞ ¼ 1 can be found in [13], p. 301–302. Note that the Schwartz

function considered there differs from ours by a factor of 1=2. &

4.2.2(B). The Hyperbolic Case

So let X 2 VðRÞ such that qðXÞ < 0, say qðXÞ ¼ �d with d 2 Rþ. The stabilizer of

X1ðdÞ :¼

ffiffiffid

p0

0 �ffiffiffid

p

� �in GðRÞ ¼ SL2ðRÞ

is

GX1ðdÞðRÞ ¼a 00 a�1

� �: a 2 R�

:

So GX is conjugate to a discrete subgroup G0X of GX1ðdÞ, hence either infinite cyclic or

trivial.

Case (B1). With the above notation assume that GX is infinite cyclic, say

g � X ¼ X1ðdÞ and gGXg�1 ¼ G0

X ¼� E 0

0 E�1

�with some E > 1 (can assume E > 0,

since �I acts trivially on D).

PROPOSITION 4.5 (Theorem 3.4(v)). Let X 2 VðRÞ such that qðXÞ < 0 and GX

in¢nite cyclic.ThenXg2GXnG

jg�j0ðX; zÞj 2 L1ðGnDÞ;

unfolding in ð4:4Þ is valid andZGnD

Xg2GXnG

g�j0ðX; zÞ ¼

ZGXnD

j0ðX; zÞ ¼ 0: ð4:8Þ

Remark 4.6. This orbital integral also appears in the compact quotient case. Since

in that case all negative Fourier coefficients vanish, the above proposition also fol-

lows from the work of Kudla and Millson. However, as they only sketch the

argument in this particular case ([14], p. 138), it seems desirable to give a direct

proof.

Proof. We haveZGnD

Xg2GXnG

g�j0ðX; zÞ ¼

ZgGg�1nD

Xg2GXnG

j0ðg � X; ggg�1zÞ

¼

ZG0nD

Xg2G0

XnG0

g�j0ðX1ðdÞ; zÞ ð4:9Þ

308 JENS FUNKE



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where G0 :¼ gGg�1. We will now show the validity of the unfolding which then

proves the existence of the original integral as well.ZG0nD

Xg2G0

XnG0

g�j0ðX1ðdÞ; zÞ ¼

ZG0XnD

j0ðX1ðdÞ; zÞ

¼ e�4pdZG0XnD

4dx2

y2�

1

2p

� �e�4pd x2

y2dxdy

y2: ð4:10Þ

� E 0

0 E�1

�acts on z 2 D as z ! E2z. So a fundamental domain F of G0

XnD is the

domain bounded by the semi arcs jzj ¼ 1 and jzj ¼ E2 > 1 in the upper half plane:

F ¼ z 2 D : 14 jzj < E2� �

: ð4:11Þ

But in this domain j0ðX1ðdÞ; zÞ is clearly rapidly decreasing as z approaches the

boundary of D. So all considered integrals actually exist and unfolding is allowed.

Changing to polar coordinates we computeZF

4dx2

y2�

1

2p

� �e�4pdx

2

y2dxdy

y2

¼

Z E2

1

Z p

0

4d cot2ðyÞ �1

2p

� �e�4pd cot2ðyÞ 1

rcsc2ðyÞdydr

¼ logðE2ÞZ 1

�1

4pdt2 �1

2p

� �e�4pdt2dt ðt ¼ cot yÞ

¼ 0; ð4:12Þ

since Zt2 �

1

2p

� �e�pt2dt ¼

1

2pte�pt2 þ C: ð4:13Þ

&Case (B2).

PROPOSITION 4.7 (Theorem 3.4(v)). Let qðXÞ be negative and assume that GX istrivial.ThenX

g2G

g�j0ðX; zÞ 2 L1ðGnDÞ ð4:14Þ

and ZGnD

Xg2G

g�j0ðX; zÞ ¼1


p Z 1

1

t�3=2e4pqðXÞtdt: ð4:15Þ

Proof. Let qðXÞ ¼ �d and suppose that GX is trivial. Recall that we chose g 2 GðRÞ

such that

g � X ¼ X1ðdÞ ¼

ffiffiffid

p0

0 �ffiffiffid

p

� �and gGg�1 ¼ G0:




https://doi.org/10.1023/A:1020002121978


As above we haveZGnD

Xg2G

g�j0ðX; zÞ ¼

ZG0nD

Xg2G0

g�j0ðX1ðdÞ; zÞ: ð4:16Þ

Unfolding in this situation will be not possible since

j0ðX1ðdÞ; zÞ ¼ e�4pd 4dx2

y2�

1

2p

� �e�4pdx

2

y2dxdy

y2ð4:17Þ

is not integrable over D. So we have to proceed more carefully. GX is trivial. Hence

X?, having signature ð1; 1Þ, is split. Therefore the stabilizers in G of the isotropic lines

in X? are nontrivial by Lemma 4.2. By conjugation we consider the isotropic lines in

X1ðdÞ?. They are generated by

X0 ¼0 10 0

� �and fX0 ¼

0 0�1 0

� �:

We put J ¼�

0 1

�1 0

�. Note that J switches the isotropic lines (i.e. the cusps) orthogonal

to X1ðdÞ: JX0 ¼ fX0. Hence, G0X0

¼ hTai with Ta ¼�1 a0 1

�for some a 2 Rþ and

G0eX0

¼ hJTbJ�1i for some b 2 Rþ. We write G00

X0¼ hTbi. Note that a and b are not

intrinsic to the situation since they depend on the choice of g such that

g � X ¼ X1ðdÞ. Define a subset G of D ¼ H by

G ¼ fz 2 D : jzj5 1g: ð4:18Þ

Note JG ¼ �G :¼ D � G (up to measure zero). Fundamental for us is the fact that J

essentially fixes X1ðdÞ: J:X1ðdÞ ¼ �X1ðdÞ; hence, for any z 2 D,

j0ðX1ðdÞ; zÞ ¼ j0ðJ:X1ðdÞ; zÞ ð4:19Þ

and therefore (up to measure zero)

j0ðX1ðdÞ; zÞ ¼ wGðzÞj0ðX1ðdÞ; zÞ þ J � wGðzÞj

0ðX1ðdÞ; zÞ� �

: ð4:20Þ

Here wG denotes the characteristic function of G. We haveZGnD

��Xg2G

g�j0ðX; zÞ��

¼

ZG0nD

��Xg2G

wG ðgzÞj0ðX1ðdÞ; gzÞ þ wGðJgzÞj0ðX1ðdÞ; JgzÞ

��4ZG0nD

Xg2G0

X0nG0

��Xk2Z

wGðTka gzÞj

0ðX1ðdÞ;Tka gzÞ

��þþ

ZG0nD

Xg2G0eX0 nG0

��Xk2Z

wGðJJTkb J

�1gzÞj0ðX1ðdÞ; JJTkbJ

�1gzÞ��: ð4:21Þ

310 JENS FUNKE



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Now we are in position to unfold and obtain

¼

ZG0X0

nD

��Xk2Z

wGðTka zÞj

0ðX1ðdÞ;Tka zÞ��þ

þ

ZG0eX0 nD

��Xk2Z

wGðTkb J

�1zÞj0ðX1ðdÞ;TkbJ

�1zÞ��

¼

ZG0X0

nD

��Xk2Z

wGðTka zÞj

0ðX1ðdÞ;Tka zÞ��þ

þ

ZG00X0

nD

��Xk2Z

wGðTkb zÞj

0ðX1ðdÞ;Tkb zÞ�� ð4:22Þ

by changing variables z j�! Jz in the second integral. It is sufficient to show

the convergence of the first integral. So let F1 be a fundamental domain for

G0X0

nD and split F1 ¼ F 1 q F 2, where F 1 ¼ z 2 F1 : ImðzÞ5 1� �

. For F 1, we

have ZF 1

��Xk2Z

wGðTkazÞj

0ðX1ðdÞ;TkazÞ��

¼

ZF 1

��Xk2Z

j0ðX1ðdÞ; z þ akÞ��

¼ e�4pdZF 1

��Xk2Z

4dðx þ akÞ2

y2�

1

2p

� �e�4pd ðxþakÞ2

y2

�� dxdyy2

¼1

8d3=2ae�4pd

ZF 1

�� Xw2a�1Z

y3w2e�py2w2

e2pixw�� dxdy

y2ð4:23Þ

by Poisson summation (see Section’4.1.). But the integrand is clearly exponentially

decreasing for y ! 1. This shows the existence of this part of the integral. M More-

over, removing the absolute value signs, we see that the integral (4.23) vanishes, as

we easily conclude by interchanging the summation and the integration w.r.t. x in

the last line of (4.23). However, note that

Xk2Z

4dðx þ akÞ2

y2�

1

2p

� �e�4pd ðxþakÞ2

y2dxdy

y2ð4:24Þ

is not termwise integrable over F 1.

For F 2, we can unfold even further and getZfz2G:y4 1g

j0ðX1ðdÞ; zÞ�� ¼ e�4pd

Zfz2G:y4 1g

�� 4dx2

y2�

1

2p

� �e�4pdx

2

y2

�� dxdyy2

ð4:25Þ

which is clearly finite since the integrand is rapidly decreasing at the boundary of the

domain of integration. Note that the last expression does not depend on a. This




https://doi.org/10.1023/A:1020002121978


shows integrability for the first summand in (4.22); the second summand is handled

in the same manner! These considerations give usZGnD

Xg2G

g�j0ðX; zÞ ¼ 2

Zfz2G:y4 1g

j0ðX1ðdÞ; zÞ ð4:26Þ

¼ 2e�4pdZ

fz2G:y4 1g

4dx2

y2�

1

2p

� �e�4pdx

2

y2dxdy

y2ð4:27Þ

Using (4.13) we get

2

Zfz2G:y4 1g

j0ðX1ðdÞ; zÞ

¼ 2

Z 1

0

2

Z 1ffiffiffiffiffiffiffiffi1�y2

p e�4pd 4dx2

y2�

1

2p

� �e�4pdx

2

y2dxdy

y2

¼ 4e�4pdZ 1

0

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 � y2

p2p

e�4pd1�y2

y2 y�2dy

¼1

p

Z 1

1

ffiffiffiffiffiffiffiffiffiffiffiffiw � 1

pe�4pdww�1dw ðw ¼ y�2Þ

¼1

p

Z 1

0

w12ðw þ 1Þ�1e�4pdðwþ1Þdw

¼e�4pd

2ffiffiffip

p C3

2;3

2; 4pd

� �;

where Cða; g; zÞ is the confluent hypergeometric function of the second kind which

has for ReðaÞ;ReðzÞ > 0 the integral representation

Cða; g; zÞ ¼1

GðaÞ

Z 1

0

ta�1ðt þ 1Þg�a�1e�ztdt ð4:29Þ

([16], 9.11.6). Further note the functional equation

Cða; g; zÞ ¼ z1�gCð1 þ a � g; 2 � g; zÞ ð4:30Þ

for jargðzÞj < p ([16], 9.10.8). Thus

e�4pd

2ffiffiffip

p C3

2;3

2; 4pd

� �¼

1

4ffiffiffid

ppe�4pd C 1;

1

2; 4pd

� �¼

1

4ffiffiffid

pp

Z 1

0

ðt þ 1Þ�3=2e�4pdðtþ1Þdt

¼1

4ffiffiffid

pp

Z 1

1

t�3=2e�4pdtdt: ð4:31Þ

This proves the proposition. &

312 JENS FUNKE



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4.2.3(C). The Parabolic Case

Now let ‘ be an isotropic line in VðRÞ such that its (pointwise) stabilizer G‘ is non-

trivial. Pick a point Y 2 ‘. Choose g 2 GðRÞ ¼ SL2ðRÞ such that g:Y ¼ bX0 with

X0 ¼�0 1

0 0

�and b 2 R�. Put G0 ¼ gGg�1. Hence G0

X0:¼ gGYg

�1 is equal to

f��1 ka0 1

�: k 2 Zg (if �I 2 G) for some a 2 Rþ. Recall that we defined the width of

the cusp ðY;GÞ by EðY;GÞ ¼ a=jbj ¼ EðY;GÞ.

PROPOSITION 4.8 (Theorem 3.4(iv)). Let ‘ 2 VðRÞ isotropic be a cusp of G asabove,Y 2 ‘, hY 2 QY and r 2 R�.ThenZ

GnD

XX2GðZYþhYÞ

X6¼0

j0ðrX; zÞ ¼EðY;GÞ

2pjrjð4:32Þ

where EðY;GÞ is the width of the cusp ðY;GÞ ðDef. 3:2Þ.

Proof. We can assume b ¼ �1 and by abuse of notation hY ¼ h ¼ hX0 with

h 2 ½0; 1Þ. We are interested inZGnD

XX2GðZYþhÞ

X 6¼0

j0ðX; zÞ

¼

ZGnD

Xg2GYnG

X1k¼�1

0

g�j0ðkY þ h; zÞ

¼

ZgGg�1nD

Xg2GYnG

X1k¼�1

0

j0ðg � ðkY þ hÞ; ggg�1zÞ

¼

ZG0nD

Xg2G0

X0nG0

X1k¼�1

0

g�j0ð�kX0 þ h; zÞ; ð4:33Þ

whereP01

k¼�1 omits k ¼ 0 in the case of the trivial coset. Now

j0ð�rðk þ hÞX0Þ; zÞ ¼ððk þ hÞrÞ2

y2�

1

2p

� �e�pððkþhÞrÞ2

y2dxdy

y2: ð4:34Þ

We unfold and getZG0nD

Xg2G0

X0nG0

X1k¼�1

0

g�j0ð�rðk þ hÞX0; zÞ

¼

ZG0X0

nD

X1k¼�1

0 ððk þ hÞrÞ2

y2�

1

2p


y2dxdy

y2

¼ EðY;GÞ

Z 1

0

X1k¼�1

0 ððk þ hÞrÞ2

y2�

1

2p


y2 y�2dy: ð4:35Þ




https://doi.org/10.1023/A:1020002121978


The validity of the unfolding in (4.35) (and therefore the existence of the original

integral) follows by looking at the last integral: For y ! 0 we have exponential

decay and for y ! 1 one sees-adding the constant term k ¼ 0 into the summation

if h ¼ 0–by Poisson summation (see Section 4.1.)

X1k¼�1

0 ðkrÞ2

y2�

1

2p

� �e�pðkrÞ2

y2dxdy

y2

¼1

2p� r�1y3

X1w¼�1

ðw=rÞ2e�pðyw=rÞ2 !

dxdy

y2ð4:36Þ

which is Oðy�2Þ as y ! 1. (The same argument works for h 6¼ 0.) Now in the last

expression of (4.35) interchanging summation and integration is not allowed in gen-

eral since

X1k¼�1

0 ðkrÞ2

y2e�pðkrÞ2

y2 ¼X1

k¼�1

0 1

2pe�pðkrÞ2

y2 ¼ OðyÞ ðy ! 1Þ ð4:37Þ

However, we can modify the integrand defining

FðsÞ :¼

Z 1

0

X1k¼�1

0 ððk þ hÞrÞ2

y2�

1

2p


y2 y�2�sdy: ð4:38Þ

for s 2 C. F is holomorphic for ReðsÞ > �1 and for ReðsÞ > 0 interchanging summa-

tion and integration is valid. We obtain

FðsÞ ¼X1

k¼�1

0Z 1

0

ððk þ hÞrÞ2

y2�

1

2p


y2 y�2�sdy ð4:39Þ

¼ 12 p

�3�s2

P1

k¼�1

0

jðk þ hÞrj�1�s�

�

Z 1

0

w �1

2

� �wðsþ1Þ=2

ewdw

ww ¼

pðk þ hÞr2

y2

� �ð4:40Þ

¼ 12 p

�3�s2 jrj�1�s zð1 þ s; hÞ þ zð1 þ s; 1 � hÞð Þ�

� Gs þ 3

2

� ��1

2G

s þ 1

2

� �� ð4:41Þ

¼ 12 p

�3�s2 jrj�1�s zð1 þ s; hÞ þ zð1 þ s; 1 � hÞð Þ�

�s

2G

s þ 1

2

� �!

1

2pjrj; ð4:42Þ

as s ! 0. Here zðs; xÞ ¼P1

n¼0ðx þ nÞ�s is the Hurwitz zeta function which has a

simple pole of residue 1 at s ¼ 1.

314 JENS FUNKE



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5. General Signature ðp; 2Þ

5.1. THETA INTEGRAL FOR SOðp; 2Þ

Now we assume that V has signature ðp; 2Þ. Let U V be a positive definite subspace

of dimension p � 1 so that CU ¼ GUnDU is a quotient of H in M. We consider the

period integral

IjVðt;L;CUÞ ¼

ZCU

yjVðt;LÞ: ð5:1Þ

Here we write jV for j to emphasize the domain j is associated to. For p ¼ 1 and

U ¼ ð0Þ, this is the theta integral considered in the previous sections. We write

LU ¼ L \ U and LU? ¼ L \ U?: ð5:2Þ

We can split the lattice L as follows:

L ¼Xri¼1

ðli þ LUÞ ? ðmi þ LU?Þ ð5:3Þ

with li 2 L#U and mi 2 L#

U? .

THEOREM 5.1.

IjVðt;L;CUÞ ¼

Xri¼1

Wðt; li þ LUÞIjU?ðt; mi þ LU?Þ:

Here Wðt; li þ LUÞ ¼P

x2liþLUe2piqðxÞt is the standard theta function ofa cosetof the posi-

tive de¢nite latticeLU. In particular, the period integral IjVðt;L;CUÞ is a nonholomorphic

(ifCU is noncompact) modular form ofweight ðp þ 2Þ=2.Proof. By Theorem 2.1 (ii), we have that under the pullback i�U : O1;1

ðDÞ !

O1;1ðDUÞ of differential forms,

i�UjV ¼ jþU � jU? ; ð5:4Þ

where jþU is just the Gaussian on U. Then

yjVðt;LÞ ¼

Xx2L

jVðX; tÞ ð5:5Þ

¼Xri¼1

Xx2liþLU

jþUðx; tÞ

Xy2miþLU?

jU?ðy; tÞ: ð5:6Þ

integrating over CU together with the results on the theta integral for signature ð1; 2Þ

(Theorem 3.4) now gives the theorem. &

5.2. INTERSECTION NUMBERS

We will now show how one can interpret the Fourier coefficients of yjVðt;LÞ as inter-

section numbers. For a special curve C ¼ CU (dimU ¼ p � 1) and a divisor C0 ¼ CU0

(dimU0 ¼ 1), we define the intersection number in (the interior of) M:




https://doi.org/10.1023/A:1020002121978


½C � C0�M :¼ ½C � C0�tr

þ volðC \ C0Þ1; ð5:7Þ

the sum of the transversal intersection and the (hyperbolic) volume of the one-

dimensional intersection of C and C0. For p ¼ 2, the Hilbert modular surface case,

this follows Hirzebruch and Zagier ([7]). One easily sees that CU and CU0 intersect

transversally if and only if U þ U0 is positive definite of (maximal) dimension p, while

CU and CU0 have a one-dimensional intersection if and only if CU # CU0 , i.e., U0 # U.

THEOREM 5.2. Assume for simplicity that L ¼ LU þ LU?. Also assume that G is tor-sion-free so that M has no quotient singularities. For a composite special divisor CN

ðN 2 NÞ, we have

ðiÞ ½CU � CN�tr

¼P

N1 5 0;N2>0N1þN2¼N

rðN1;LUÞ degðN2;CUÞ;

ðiiÞ volðCU \ CNÞ1¼ rðN;LUÞvolðCUÞ:

Here

rðN;LUÞ ¼ #fx 2 LU : qðxÞ ¼ Ng

and

degðN;CUÞ ¼ #GUnfx 2 LU? : qðxÞ ¼ Ng;

the degree of the Heegnerdivisor in the modularcurveCU.Proof. We write CN ¼

Px2GnLN

Cx. We decompose x ¼ x1 þ x2 with x1 2 LU and

x2 2 LU? . We have CU Cx if and only if x2 ¼ 0. In the sum defining CN therefore

exactly the x ¼ x1 2 LU of length N contribute to the one-dimensional intersection.

This proves (ii). For (i), each x with qðx2Þ ¼ N2 > 0 contributes. Taking into account

the action of GU on LU? gives the assertion. &

We certainly have a similar theorem if L does not split along U. If G is not torsion-

free, then we can always pass to a torsion-free subgroup G0 of finite index to obtain

intersection numbers on G0nD. The intersection numbers on GnD (in the sense of

rational homology manifolds) are then obtained by dividing by the degree of the

covering G0nD j�!GnD.

COROLLARY 5.3. Write

IjVðt;L;CUÞ ¼

X1N¼0

cðNÞqN þX1

N¼�1

cðN; vÞqN

for the Fourier expansion of the above theta integral.Then

cðNÞ ¼ ½CU � CN�M;

i.e., the intersection numbers are exactly the Fouriercoe⁄cients of the holomorphic partofIjV

ðt;L;CUÞ.Proof. Just write down the Fourier expansion of IjV

ðt;L;CUÞ using Theorem 3.5

and 5.1. &

316 JENS FUNKE



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5.3. HIRZEBRUCH–ZAGIER CASE

As an example we will derive the basic case of the results of Hirzebruch and

Zagier [7] and its (mild) extensions by Franke [5], Hausmann [6] and van der

Geer [23].

Let D > 0 be the discriminant of the real quadratic field K ¼ QðffiffiffiffiD

pÞ over Q, OK

its ring of integers. We denote by x j�!x0 the Galois involution on K. We let

V M2ðKÞ be the space of skew-Hermitian matrices in M2ðKÞ, i.e., which satisfy

the relation tX0 ¼ �X. V is a vector space over Q of dimension 4. We let L V

be the integral skew-Hermitian matrices; that is

L ¼ X ¼affiffiffiffiD

pl

�l0 bffiffiffiffiD

p

� �: a; b 2 Z; l 2 OK

: ð5:8Þ

As usual, the determinant defines a quadratic form on L; we have QðXÞ ¼ abD þ ll0,

which is of signature ð2; 2Þ and has Q-rank 1, i.e., it splits over Q into a hyperbolic

plane and into an anisotropic part of rank 2. SL2ðOKÞ acts on L by g � X ¼ gXtg0 asisometries. We let G be the Hurwitz–Maass extension of SL2ðOKÞ which is the maxi-

mal discrete subgroup of PGLþ2 ðRÞ

2 containing SL2ðOKÞ (for a brief discussion of its

definition and properties, see [23]; for example, if D � 1mod 4 is a prime, then

G ¼ SL2ðOKÞ . This is actually the case Hirzebruch and Zagier considered). Slightly

changing our notation we write S ¼ GnD for the Hilbert modular surface. The

Hirzebruch–Zagier cycles TN are nothing but our special cycles CN ¼P

X2LNCX.

TN has finitely many components, is nonempty if ðN=pÞ 6¼ �1 and is compact if N

is not the norm of an ideal in OK. We can compactify S by adding the cusps and

resolving the singularities thus created. We call this (up to quotient singularities)

nonsingular compact surface ~S. Now H2ð ~SÞ decomposes canonically into as the

direct sum of the image of H2ðSÞ and the subspace generated by the homology cycles

of the curves of the cusp resolution. We denote by TcN the component in the first fac-

tor of TN. Hirzebruch and Zagier compute the intersection numbers ½TcN � TM�

(N;M 2 N) and by a direct computation they show that these numbers are the Four-

ier coefficients of a holomorphic modular form of weight 2. In fact, ½TcN � TM� is the

sum of the intersection numbers ½TN � TM�S in the interior and the contribution of

the cusp resolution, and these numbers individually occur as Fourier coefficients

of two nonholomorphic modular forms. For M ¼ 1, one has

ðT1 � TNÞS ¼ HDðNÞ ¼X

s244Ns2�4NmodD

H4N � s2

D

� �; ð5:9Þ

where HðNÞ denotes the class number of positive definite binary quadratic forms, see

Example 3.9. We now recover these results: We consider the curve T1 ¼ C1 in our

setting. G acts transitively on L1, the vectors of length 1, see [6,23]. So T1 ¼ CX0

for any X0 2 L1. We pick X0 ¼�

0 1

�1 0

�2 L.




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THEOREM 5.4.

1

2

ZT1

yjVðt;LÞ ¼

X1N¼0

HDðNÞqN þ2v�1=2ffiffiffiffi

Dp

Xl2O

b 4pl � l0

2

� �2

v

!qll

0

:

So the Fourier coe⁄cients of the holomorphic part of the period integralRT1yjV

ðt;LÞ arethe intersection numbers of the cyclesT1 andTN in the‘interior’ofGnD.

Proof. First we assume D � 0mod 4, so K ¼ Qðffiffiffid

pÞ with d ¼ D=4 square free. In

this case L splits orthogonally:

L ¼ ZX0 ? ðL \ X?0 Þ ð5:10Þ

and

LU? ¼ L \ X?0 ¼

ffiffiffid

p 2a b�b 2c

� �: a; b; c 2 Z

; ð5:11Þ

which is—up to the scaling—exactly the lattice considered in Example 3.9(i) which

gave rise to Zagier’s Eisenstein series F !. Hence by Theorem 5.1 and Example

3.9(i) we find

12Ijðt;L;T1Þ ¼ WðtÞIjU?

ðt;LU?Þ ð5:12Þ

¼ WðtÞF ðdtÞ ð5:13Þ

¼Xm2Z

qm2

!�

�X1N¼0

HðNÞqdN þ ðdvÞ�1=2Xn2Z

bð4pn2dvÞ q�dn2

!ð5:14Þ

¼Xs2 4N

s2�NðdÞ

HN � s2

d

� �qNþ

þv�1=2ffiffiffi

dp

X1N¼�1

Xn;m2Z

m2�dn2¼N

bð4pdn2vÞ qN ð5:15Þ

¼X

ð2sÞ2 4 4N

ð2sÞ2�4N ð4dÞ

H4N � ð2sÞ2

4d

� �qNþ

þ2v�1=2ffiffiffiffiffi

4dp

Xl2O

b 4pl � l0

2

� �2

v

!qll

0

ð5:16Þ

318 JENS FUNKE



https://doi.org/10.1023/A:1020002121978


since O ¼ fm þ nffiffiffid

p: m; n 2 Zg. Noting D ¼ 4d, the theorem follows for

D � 0mod 4. The case D � 1mod 4 is slightly more complicated since L does not

split orthogonally so that one has to deal with cosets. We have

LU? ¼ L \ X?0 ¼

ffiffiffiffiD

p a b�b c

� �: a; b; c 2 Z

ð5:17Þ

which is the lattice from Example 3.9(ii)! We set X1 ¼�

0ffiffiffiD

pffiffiffiD

p0

�and obtain

L ¼ ZX0 ? LU?ð Þ $ 12ðX0 þ X1Þ þ ZX0 ? LU?ð Þ: ð5:18Þ

The holomorphic part is given byX1j¼0

Xm2Z

q mþj2ð Þ

2

! X1n¼0

Hð4n � jÞqD n� j4ð Þ

!¼X1j¼0

X1N¼0

Xm;n2Z

�Hð4n � jÞqN; ð5:19Þ

where the inner summation extends over all integers m and n such that

m þj

2

� �2

þD n �j

4

� �¼ N

or, equivalently,

4n � j ¼4N � ð2m þ jÞ2

D:

So (5.19) becomesX1j¼0

X1N¼0

Xð2mþjÞ2 4 4N

ð2mþjÞ2�4N ðDÞ

H4N � ð2m þ jÞ2

D

� �qN

¼X1N¼0

Xs2 4 4N

s2�4N ðDÞ

H4N � s2

D

� �qN: ð5:20Þ

For the nonholomorphic part, we first note

O ¼ m þ n1 þ

ffiffiffiffiD

p

2: m; n 2 Z

:

For l ¼ m þ n1 þ

ffiffiffiffiD

p

2we write l % ðm; nÞ. Then

X1j¼0

Xm2Z

q mþj2ð Þ

2

!2ðvDÞ

�1=2Xn2Z

b 4p n þj

2

� �2

vD

!q�D nþ j

2ð Þ2

!ð5:21Þ

¼2v�1=2ffiffiffiffi

Dp

X1j¼0

X1N¼0

Xm;n2Z

�b 4p n þj

2

� �2

Dv

!qN; ð5:22Þ




https://doi.org/10.1023/A:1020002121978


where the inner sum goes over all m; n such that ðm þ ðj=2ÞÞ2 � Dðn � ðj=4ÞÞ ¼ N.


Dp

X1j¼0

XN2Z

Xl%ðn�m;2nþjÞ

ll0¼N

b 4pl � l0

2

� �2

v

!qN ð5:23Þ


Dp

Xl2O

b 4pl � l0

2

� �2

v

!qll

0

; ð5:24Þ

as desired. &

The factor 12 in the previous theorem occurs as �1 2 G, which acts trivially on

D ’ H � H. To obtain the intersection numbers ½Tc1:TN� and the holomorphic modu-

lar form of weight 2, one can now apply the holomorphic projection principle ontoRT1yjV

ðt;LÞ. This is an idea of van der Geer and (independently) Zagier, which has

been carried out in [23].

Remark 5.5. We see that Theorem 5.1 is the exact generalization of a part of the

results of Hirzebruch–Zagier. One is certainly very interested to obtain complete

analogues of these results. Applying holomorphic projection onto the functionRCU

yjVðt;LÞ in Theorem 5.1 for p > 2 is certainly possible, but this does not have a

priori a geometric interpretation. More promising seems to be an analysis of the

boundary along the lines of the theory of Kudla and Millson. It seems likely that

Eisenstein cohomology will enter the picture at this stage. We hope to come back to

this issue.

Acknowledgements

Most of the results in this paper are part of the author’s PhD thesis at the University

of Maryland. I would like to thank my advisor Steve Kudla for his guidance

throughout the last years. I also would like to thank John Millson for many valuable

discussions. The final stages of this paper were completed while visiting the Max-

Planck-Institut f �ur Mathematik in Bonn as a fellow of the European PostDoc Insti-

tute (EPDI). I would like to thank its directors in general for providing such a stimu-

lating environment and in particular Professor Zagier for some comments on the

results of this paper.

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https://doi.org/10.1023/A:1020002121978


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Heegner Divisors and Nonholomorphic Modular Forms€¦ · modular surface. As an example for Theorem 1.2we then explicitly derive these parts. From there one can obtain the complete